A bag contains $5$ red balls, $3$ green balls, and $3$ blue balls. If we choose a ball, then another ball without putting the first one back in the bag, what is the probability that the first ball will be green and the second will be red? Write your answer as a simplified fraction.
The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened In this case, event A is picking a green ball and leaving it out. Event B is picking a red ball. Let's take the events one at at time. What is the probability that the first ball chosen will be green? There are $3$ green balls, and $11$ total, so the probability we will pick a green ball is $\dfrac{3} {11}$ After we take out the first ball, we don't put it back in, so there are only $10$ balls left. Since the first ball was green, there are still $5$ red balls left. So, the probability of picking a red ball after taking out a green ball is $\dfrac{5} {10}$ Therefore, the probability of picking a green ball, then a red ball is $\dfrac{3}{11} \cdot \dfrac{5}{10} = \dfrac{3}{22}$